Question: Simplify and expand the following expression: $ \dfrac{3}{p - 6}- \dfrac{2}{4p + 40}- \dfrac{4}{p^2 + 4p - 60} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the second term: $ \dfrac{2}{4p + 40} = \dfrac{2}{4(p + 10)}$ We can factor the quadratic in the third term: $ \dfrac{4}{p^2 + 4p - 60} = \dfrac{4}{(p - 6)(p + 10)}$ Now we have: $ \dfrac{3}{p - 6}- \dfrac{2}{4(p + 10)}- \dfrac{4}{(p - 6)(p + 10)} $ The least common multiple of the denominators is: $ (p - 6)(p + 10)$ In order to get the first term over $(p - 6)(p + 10)$ , multiply by $\dfrac{4(p + 10)}{4(p + 10)}$ $ \dfrac{3}{p - 6} \times \dfrac{4(p + 10)}{4(p + 10)} = \dfrac{12(p + 10)}{(p - 6)(p + 10)} $ In order to get the second term over $(p - 6)(p + 10)$ , multiply by $\dfrac{p - 6}{p - 6}$ $ \dfrac{2}{4(p + 10)} \times \dfrac{p - 6}{p - 6} = \dfrac{2(p - 6)}{(p - 6)(p + 10)} $ In order to get the third term over $(p - 6)(p + 10)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{4}{(p - 6)(p + 10)} \times \dfrac{4}{4} = \dfrac{16}{(p - 6)(p + 10)} $ Now we have: $ \dfrac{12(p + 10)}{(p - 6)(p + 10)} - \dfrac{2(p - 6)}{(p - 6)(p + 10)} - \dfrac{16}{(p - 6)(p + 10)} $ $ = \dfrac{ 12(p + 10) - 2(p - 6) - 16} {(p - 6)(p + 10)} $ Expand: $ = \dfrac{12p + 120 - 2p + 12 - 16}{4p^2 + 16p - 240} $ $ = \dfrac{10p + 116}{4p^2 + 16p - 240}$ Simplify: $ = \dfrac{5p + 58}{2p^2 + 8p - 120}$